DK-GEORGMOHR 2008

Denmark has played an international football match against Georgia. the fight ended $5-5$ , and between the first and the last goal the game has justnever stood . No country has scored three goals in a row, and Denmark scored the sixth goal. Can you use this information to determine which country scored the fifth goal?

If three integers $p, q$ and $r$ apply that $$p + q^2 = r ^2.$$ Show that $6$ adds up to $pqr$ .

The numbers from $1$ to $500$ are written on the board. Two players $A$ and $B$ erase alternately one number at a time, and $A$ deletes the first number. If the sum of the last two number on the board is divisible by $3$ , $B$ wins, otherwise $A$ wins. Which player can lay out a strategy that ensures this player's victory?

In triangle $ABC$ we have $AB = 2, AC = 6$ and $\angle A = 120^o$ . The bisector of angle $A$ intersects the side BC at the point $D$ . Determine the length of $AD$ . The answer must be given as a fraction with integer numerator and denominator.